I have an adage that I like - it's one that I think sort of works as a nice little guide. The Christian seeks neither to give offense nor to be offended
This is actually one of my little pithy summations of what the Scriptures teach. Don't try to offend people, and don't let yourself get offended.
Now, when the Scriptures speak to "offend" they literally mean the abandonment of faith. Clearly you shouldn't be damaging people's faith, nor should you engage in activity that would destroy your own.
However, let's loosen the language here and make it more... general, more in line with how modern English speaks of being offended. Or upset. Or ticked off.
These are all different ways of saying "being angry" - but they all have something in common. They all refer, linguistically, to being thrown off balance. If you are offended, you are off - ended. You are no longer on your proper end. If you are upset you have been tipped over. And if you are ticked off, you are moved off of your proper and normal tick.
Think like an athlete for a moment. If I am going to hit a golf ball (or throw a punch), my legs are vital. The ability to drive, to use force all comes from... my balance. Try hitting a golf ball while not balanced. (Or watch someone who is off balance try to tackle someone... ugh). You can't. You lose all strength if you lose your balance.
Likewise - if I am offended, if I am upset, I have lost my balance -- and I become weak. Powerless. I cannot speak correctly and too the point... and I will miss the mark. I will fall into gross sin.
Seriously, that's why I don't trust anger -- it throws me off balance, and instead of thinking about how best I can serve my neighbor... I think of other things, vile things - and then I don't speak truly anymore.
Likewise, if I upset my neighbor, if I throw them off balance -- it makes it hard for them to show love (of course, the exception is if I have upset them off of their open sin... sometimes if our tick is the wrong tick, we need to be ticked off). Hence, if I unduly upset my neighbor, I am simply hindering them in their own tasks of showing love.
Thus - try not to give offense and don't take offense.
Oh, and don't play the "offend" card either - it's tacky and cliche'.
And just as another note -- if you are in a discussion with someone, and they brush aside your argument and land a solid counter punch... that's not a sign of them being offended. Just saying.